[FOM] Foundational Challenge

Wed Jan 22 02:30:47 EST 2020

Dear Patrick,
Thank you for your thoughtful letter.
I agree that “… this does not obviously mean that it is impossible to create one…”. 
For me a category is defined as a collection of morphisms and a collection of objects with axioms relating them.
The objects certainly do not have to be sets, but the collections are there, and I do not know how to even speak of a category without those collections in the background.
There is no question that the bottom line in some diagrammatic category (e.g. the Temperley Lieb Diagram Category) may be the structure of the diagrams and their compositions, and this
is conceptually prior to thinking of them either as point sets or graphs or looking at the collections of them. So conceptually one is not taking sets as prior but rather as descriptive and multiply so.
Not all ways of thinking of mathematics intuitively and conceptually begin or end with collections. And so we have the  "intertwining" of set theory and mathematical structures (whose concepts are wider
than a given set theoretic description). I like to start mathematical thoughts with images, geometrical diagrams, 
topological diagrams, visual associations made formal, certain distinctions … And once enough clarity has formed in such a domain, collections of entities emerge and we need to think set theoretically to continue to relate that locally grown mathematics with the rest of the world of study. Lets take a very simple example. Here is a category C with one object O and one non-identity morphism f:O——>O. Then we have the collection of all compositions {f,ff,ffff,…} and we will want to discuss whether this collection is finite or infinite. We have walked at once into considerations of elementary set theory on examining the nearly simplest category. Sets and Categories are indeed intertwined.

We would never have regarded topological spaces as morphisms in a monoidal category, as is part and parcel of TQFT and Quantum Topology, without a category theoretic point of view.
Categories make possible conceptual shifts that a strictly set theoretic point of view would not see. This is normal. Would set theory alone give rise to calculus or topology or group theory?
The point we are both making is that Category is different from specific structures like Groups or Topological Spaces because of its generality and simplicity. 
Category is foundational and belongs in the discussion of FOM.
Best,
Lou Kauffman

> On Jan 22, 2020, at 12:47 AM, Patrik Eklund <peklund at cs.umu.se> wrote:
> 
> Dear Louis,
> 
> When you say
> 
>> There is no felicitous way ...
> 
> I guess you actually say you have not been able to create one, or you haven't seen anybody else having created one. And if so, this does obviously not mean it is impossible to create one.
> 
> Let me give you another example, not related to your graphs, but what we nevertheless find as a "felicious way".
> 
> Look at signatures, i.e., sorts and operators. We usually, and in particular if we do not use categories, view them as sets of sorts and sets of operators. However, they can be arranged as objects in a monoidal category (see pages 219-221 in our Fuzzy Terms paper, https://www.sciencedirect.com/science/article/pii/S0165011413000997). This can be seen as an extensions to what Benabou did in the 1960s.
> 
> What has been said in previous FOM postings by others generally about "what we want to do with them", in this particular case is e.g. for the Goguen category to be viewed as a monoidal category, which enables algebraic many-valuedness to be attached with operators. This is an important step for the foundations of many-valued logic and as contribution to the long-standing debate between "fuzzy" and "probability". This "felicious way" of viewing signatures in a many-valued setting can be used also in very practical situations, like analysing information structures related with disease and functioning in health care.
> 
> What I say here is that we would never have seen this if we would have continued to view sorts and operators to reside in 'sets' only. So the meaning of
> 
>> The whole enterprise is embedded in set theory ...
> 
> is now very different as expected. Larry Paulson was a bit into this also when sharing his view on how category theory builds upon set theory. This is different, I guess, but nevertheless related to the "intertwining" of set theory and category theory.
> 
> Needless to say, tree automata and such things could also quite felicitously be handled similarly, and then we are a bit on the graph side, aren't we?
> 
> Best,
> 
> Patrik
> 
> 
> 
> On 2020-01-21 06:31, Louis H Kauffman wrote:
>> Dear Jose M.,
>> Lets take knot diagrams as an example. A given knot diagram D is a
>> 4-valent plane graph with extra structure. It is convenient to
>> condsider the set of  all diagrams S(D) obtained from D by the
>> Reidemeister moves and to prove that a certain knot polynomial will
>> take the same value on all of them. While one may focus on the diagram
>> or the graph, nevertheless one needs the set of all the diagrams and
>> the
>> understanding that S(D) is a countable set and other matters. There is
>> no felicitous way to handle knot diagrams and the associated tensor
>> categories and functors of quantum topology without using set theory.
>> The whole enterprise is embedded in set theory with special languages
>> of diagrams and compositionality that are convenient for both theory
>> and calculation. This is what we expect of set theory - a broad basis
>> that can handle special inventions as well. One more point. The
>> diagram D can be seen as a projection of a curve that is
>> set-theoretically embedded in three dimensional space. This
>> relationship is of key importance for
>> working with the knot theory and relating the diagrammatic parts of it
>> with the classical, topological and geometrical parts. The set theory
>> allows one to work with the subject as a whole. These remarks apply to
>> the categories for quantum theory and the tensor diagrams as well.
>> Very best,
>> Lou Kauffman
>>> On Jan 17, 2020, at 12:20 PM, José Manuel Rodríguez Caballero
>>> <josephcmac at gmail.com> wrote:
>>> Tim wrote:
>>>> So again, I don't agree with M. Katz that ZFC as a universal
>>>> foundation for all mathematics is not credible, if we understand
>>>> that
>>>> claim rightly.  The claim isn't that for every subfield X of
>>>> mathematics,
>>>> we must explicitly use raw set theory for the "foundations of X"
>>>> and
>>>> eschew any defined notions.  The claim, rather, is that set theory
>>>> is
>>>> still the most convincing candidate when it comes to Generous
>>>> Arena,
>>>> Shared Standard, and Metamathematical Corral for mathematics
>>>> considered as
>>>> a whole
>>> There is a contemporary tendency in mathematics, physics and
>>> computer science of substituting formulae by the so-called graphical
>>> calculus [1, 2, 3, 4], both in the statements of the theorems and in
>>> the proofs. In this approach, the sets are not fundamental, neither
>>> from a formal point of view nor from an intuitive point of view. The
>>> main focus is on the composition of processes and the fundamental
>>> intuition comes from topology, especially from knot theory. For this
>>> reason, this new tendency is known as compositionality [5].
>>> Compositionality cannot be reduced to the foundation of X, because X
>>> is aimed to be everything, in mathematics and outside mathematics,
>>> e.g., physicist, biology, social sciences, computer science, etc. It
>>> is not unreasonable to predict that compositionality may become
>>> someday a new Generous Arena, Shared Standard, and Metamathematical
>>> Corral for mathematics as a whole.
>>> As evidence that compositionality is already part of the current
>>> mainstream scientific activity, I would like to share the following
>>> typical fragment from an announce of Postdoctoral and Ph.D.
>>> positions in Edinburgh for a project about Quantum Theory (notice
>>> that both Category Theory and Causality, which are closely related
>>> to compositionality, are considered as important for this field):
>>>> Applicants must have or be about to receive a degree in Computer
>>>> Science, Mathematics, or Physics, with a background in one or more
>>>> of
>>>> the following areas:
>>>> * Quantum computing
>>>> * Category theory
>>>> * Programming languages
>>>> * Causality
>>>> * Concurrency
>>> Kind regards,
>>> Jose M.
>>> [1] Penrose, Roger. Applications of negative dimensional tensors.
>>> Combinatorial mathematics and its applications 1 (1971): 221-244.
>>> [2] Kauffman, Louis H. Introduction to quantum topology. _Quantum
>>> topology_. 1993. 1-77.
>>> [3] Coecke, Bob, and Aleks Kissinger. Picturing quantum processes.
>>> Cambridge University Press, 2017.
>>> [4] Blinn, Jim, Using Tensor Diagrams to Represent and Solve
>>> Geometric Problems. 2002
>>> URL =
>> https://www.microsoft.com/en-us/research/wp-content/uploads/2002/01/UsingTensorDiagrams.pdf
>>> [5] Compositionality
>>> URL = https://compositionality-journal.org/about/
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